2003/06/02 by Nicholas J. Proudfoot, Proudfoot, Nicholas J. · 2 citations
Mathematics · #52C35 #Algebraic Topology (math.AT) #Combinatorics (math.CO) #FOS: Mathematics #Rings and Algebras (math.RA) #math.AT #math.CO #math.RA #msc:52C35
paper · pdf · doi:10.48550/arxiv.math/0306013
9 pages, 2 figures. Revised exposotion, corrections to examples
arxiv created 2004/08/03 · arxiv updated 2009/11/30
Given a real arrangement A, the complement M(A) of the complexification of A admits an action of ℤ2 by complex conjugation. We define the equivariant Orlik-Solomon algebra of A to be the ℤ2-equivariant cohomology ring of M(A) with coefficients in ℤ2. We give a combinatorial presentation of this ring, and interpret it as a deformation of the ordinary Orlik-Solomon algebra into the Varchenko-Gel'fand ring of locally constant \mattbbZ2-valued functions on the complement C(A) of A in ℝn. We also show that the ℤ2-equivariant homotopy type of M(A) is determined by the oriented matroid of A. As an application, we give two examples of pairs of arrangements A and A' such that M(A) and M(A') have the same nonequivariant homotopy type, but are distinguished by the equivariant Orlik-Solomon algebra.